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EEE 431 Computational Methods in ElectrodynamicsPowerPoint Presentation

EEE 431 Computational Methods in Electrodynamics

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EEE 431 Computational Methods in Electrodynamics

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Energy and Power

- We would like to derive equations governing EM energy and power.
- Starting with Maxwell’s equation’s:

Energy and Power (Cont.)

- Apply H. to the first equation and E. to the second:

Energy and Power (Cont.)

- Subtracting:
- Since,

Energy and Power (Cont.)

- Integration over the volume of interest:

Energy and Power (Cont.)

- Applying the divergence theorem:

Energy and Power (Cont.)

- Explanation of different terms:
- Poynting Vector in
- The power flowing out of the surface S (W).

Energy and Power (Cont.)

- Dissipated Power (W)
- Supplied Power (W)

Energy and Power

- Magnetic power (W)
- Magnetic Energy.

Energy and Power (Cont.)

- Electric power (W)
- electric energy.

Energy and Power (Cont.)

- Conservation of EM Energy

Classification of EM Problems

- 1) The solution region of the problem,
- 2) The nature of the equation describing the problem,
- 3) The associated boundary conditions.

1) Classification of Solution Regions:

- Closed region, bounded, or open region, unbounded. i.e Wave propagation in a waveguide is a closed region problem where radiation from a dipole antenna is an open region problem.
- A problem also is classified in terms of the electrical, constitutive properties. We shall be concerned with simple materials here.

2)Classification of differential Equations

- Most EM problems can be written as:
- L: Operator (integral, differential, integrodifferential)
- : Excitation or source
- : Unknown function.

Classification of Differential Equations (Cont.)

- Example: Poisson’s Equation in differential form .

Classification of Differential Equations (Cont.):

- In integral form, the Poisson’s equation is of the form:

Classification of Differential Equations (Cont.):

- EM problems satisfy second order partial differential equations (PDE).
- i.e. Wave equation, Laplace’s equation.

Classification of Differential Equations (Cont.):

- In general, a two dimensional second order PDE:
- If PDE is homogeneous.
- If PDE is inhomogeneous.

Classification of Differential Equations (Cont.):

- A PDE in general can have both:
- 1) Initial values (Transient Equations)
- 2) Boundary Values (Steady state equations)

Classification of Differential Equations (Cont.):

- The L operator is now:

Classification of Differential Equations (Cont.):

- Examples:
- Elliptic PDE, Poisson’s and Laplace’s Equations:

Classification of Differential Equations (Cont.):

- For both cases a=c=1,b=0.
- An elliptic PDE usually models the closed region problems.

Classification of Differential Equations (Cont.):

- Hyperbolic PDE’s, the Wave Equation in one dimension:
- Propagation Problems (Open region problems)

Classification of Differential Equations (Cont.):

- Parabolic PDE, Heat Equation in one dimension.
- Open region problem.

Classification of Differential Equations (Cont.):

- The type of problem represented by:
- Such problems are called deterministic.
- Nondeterministic (eigenvalue) problem is represented by:
- Eigenproblems: Waveguide problems, where eigenvalues corresponds to cutoff frequencies.

3) Classification of Boundary Conditions:

- What is the problem?
- Find which satisfies within a solution region R.
- must satisfy certain conditions on Surface S, the boundary of R.
- These boundary conditions are Dirichlet and Neumann types.

Classification of Boundary Conditions (Cont.):

- 1) Dirichlet B.C.:
- vanishes on S.
- 2) Neumann B.C.:
- i.e. the normal derivative of vanishes on S.
- Mixed B.C. exits.